That was fascinating and I enjoyed it but I do have trouble with the concept of different 'sizes' of infinity although I feel sure they exist. 0r do they? What do I mean by 'exist' anyway? :-)

Infinities (it works for finites too) are compared by attempting a one to one comparison between their elements. If one can achieve this, then they are the same "size". If it is impossible to achieve this, then one of them is "bigger".

The number of errors that can be made is bigger than the number of people.
The number of ways to skin a cat is bigger than the number of cats.

There are actually an infinite number of sizes of infinity. The best way to think of an infinity is that it's the size of a set that contains all the elements. For example, the "size" (quantity of elements) that a set containing all the integers is
infinite. It is however smaller than the set of real numbers between zero and one, as shown in my original post. More generally, the quantity of subsets of an infinite set is bigger than the set being subsetted (to coin a word). (a subset is a set
which contains "some" of the elements of the original set, and no other elements, where "some" could be "all" or could be "none". Sets are designated (sometimes) by listing their elements inside curly braces; a few (i'll show three) subsets of the
days of the week are {sunday} and {tuesday, thursday, friday} and {} (that last one being the empty, or "null" set).

So, the set of positive integers ( {1,2,3,4,...} ) is not as large as the set of subsets of the positive integers ( {}, {1}, {2}, {1,2}, {1,2,3,4}, {8, 9, 423}, {500}, ... )
Note that it is perfectly fine for a set to contain sets as elements. Don't confuse a set with an element however: 1 is different from {1}. "Monday" is different from "the SET of days between Sunday and Tuesday". A car (something you can drive)
is different from "car" (the word describing something you can drive).

Ceci n'est pas une pipe.

Jose
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